Set in Discrete Topology is Clopen

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Theorem

Let $T = \struct {S, \tau}$ be a discrete topological space.

$\forall U \subseteq S: U$ is both closed and open in $\struct {S, \tau}$.


Proof

Let $U \subseteq S$.

By definition of discrete topological space, $U \in \tau$.

By definition of closed set, $\relcomp S U$ is closed in $T$, where $\relcomp S U$ is the relative complement of $U$ in $S$.

But from Set Difference is Subset:

$\relcomp S U = S \setminus U \subseteq S$

and so:

$\relcomp S U \in \tau$

That is, $\relcomp S U$ is both closed and open in $T$.

Then by Relative Complement of Relative Complement:

$\relcomp S {\relcomp S U} = U$

which is seen to be both closed and open in $T$.

$\blacksquare$


Sources

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